NONCONFORMING FINITE ELEMENTS FOR
REISSNER-MINDLIN PLATES
C. CHINOSI
Dipartimento di Scienze e Tecnologie Avanzate,Universita del Piemonte Orientale,
Via Bellini 25/G, 15100 Alessandria, ItalyE-mail: [email protected]
C. LOVADINA AND L.D. MARINI
Dipartimento di Matematica, Universita di Pavia,and IMATI–CNR,
Via Ferrata 1, 27100 Pavia, ItalyE-mails: [email protected] , [email protected]
We report on recent results about some nonconforming finite elements for plates,
introduced and analyzed in [6], [12] and [9]. All the elements are locking-free andexhibit optimal convergence rates.
Introduction
As it is well-known (cf. [4], for instance), the numerical treatment of the
Reissner-Mindlin model requires special care, in order to avoid the so-called
shear locking phenomenon and the occurrence of spurious modes.
The shear locking has its roots from the shear energy term, which for
“small” thickness enforces the Kirchhoff constraint. It turns out that for
simple low-order elements this constraint is generally too severe, thus com-
promising the quality of the obtained discrete solution. A general and
commonly adopted strategy to overcome the problem consists in modify-
ing, at the discrete level, the shear energy term, with the aim of reducing
its influence. In most cases the modification can be interpreted as the result
of (or directly arises from) a mixed approach to the problem.
However, a careless choice of the shear reduction procedure may cause a
loss of stability, typically enlighted by the presence of undesirable oscillating
components in the discrete solution (the spurious modes).
We point out that nowadays a wide choice of good elements are avalaible
1
2
in the literature (see, for instance, [1]–[3], [5]–[7], [11]–[15], and the refer-
ences therein).
The aim of this Note is to discuss some finite element schemes recently
proposed and studied in [6], [9], and [12]. These elements are all based on a
mixed nonconforming approach, and they have some features which seem to
be favorable for a possible extension to the more complex (and more inter-
esting) case of shell problems. Indeed, the methods we will consider take
advantage of low-order polynomial approximations; they are locking-free
and optimally convergent; finally, once the mixed variables (i.e. the shear
stresses) have been eliminated, all the remaining unknowns (i.e. rotations
and deflections) share the same nodes and degrees of freedom.
The paper is organized as follows. In Section 1 we recall the Reissner-
Mindlin equations, and we briefly present the elements of [6] and [12]. Sec-
tion 2 is devoted to numerical results.
1. The Reissner-Mindlin problem and the nonconforming
elements
The Reissner-Mindlin equations for a clamped plate require to find (θ, w,γ)
such that
− div C ε(θ) − γ = 0 in Ω, (1)
− div γ = g in Ω, (2)
γ = λt−2(∇w − θ) in Ω, (3)
θ = 0, w = 0 on ∂Ω. (4)
In (1)-(3), Ω is the midplane, regular and bounded, t is the thickness, C is
the tensor of bending moduli, and λ is the shear modulus (incorporating
also the shear correction factor). Moreover, θ represents the rotations,
w the transversal displacement, γ the scaled shear stresses and g a given
transversal load. Finally, ε is the usual symmetric gradient operator. The
classical variational formulation of problem (1)–(4) is
Find (θ, w,γ) ∈ H10(Ω) × H1
0 (Ω) × L2(Ω) :
a(θ,η) + (∇v − η,γ) = (g, v) (η, v) ∈ H10(Ω) × H1
0 (Ω),
(∇w − θ, τ ) − λ−1t2(γ, τ ) = 0 τ ∈ L2(Ω),(5)
3
where (·, ·) is the inner-product in L2(Ω) (or in L2(Ω)), and
a(θ,η) :=
∫
Ω
C ε(θ) : ε(η) dx. (6)
We now introduce nonconforming finite element approximations of prob-
lem (1)–(4) using the approach detailed in [6]. Let then Th be a regular
decomposition of Ω into triangular elements T (see [10], for example), and
let us set H1(Th) :=∏
T∈ThH1(T ). Following [1], we define suitable jump
operators. Let Eh denote the set of all the edges in Th, and E inh the set of
internal edges. Let e be an internal edge of Th, shared by two elements T+
and T−, and let ϕ denote a function in H1(Th), or a vector in H1(Th). For
a scalar function ϕ ∈ H1(Th) we define its jump as
[ϕ] = ϕ+n+ + ϕ−n− ∀e ∈ E inh , (7)
while the jump of a vector ϕ ∈ H1(Th) is given by
[ϕ] = (ϕ+ ⊗ n+)S + (ϕ− ⊗ n−)S ∀e ∈ E inh , (8)
where (ϕ ⊗ n)S denotes the symmetric part of the tensor product, and
n+ (resp. n−) is the outward unit normal to ∂T + (resp. to ∂T−). On
the boundary edges we define jumps of scalars as [ϕ] = ϕn, and jumps
of vectors as [ϕ] = (ϕ ⊗ n)S , where n is the outward unit normal to ∂Ω.
Following the ideas of [6], we now select finite element spaces Θh ⊂ H1(Th),
Wh ⊂ H1(Th), and Γh ⊂ L2(Ω), with the property: ∇hWh ⊆ Γh , where
∇h denotes the gradient operator element by element. The discrete problem
is then
Find (θh, wh,γh) ∈ Θh × Wh × Γh :
ah(θh,ηh) + (γh,∇hvh − Rhηh) = (g, vh) (ηh, vh) ∈ Θh × Wh,
(∇hwh − Rhθh, τh) − λ−1t2(γh, τh) = 0 τh ∈ Γh.(9)
Above, the bilinear form ah(·, ·) is defined, for piecewise regular functions,
by
ah(θ,η) :=∑
T∈Th
∫
T
C ε(θ) : ε(η) dx + pΘ(θ,η), (10)
where pΘ is a penalty term given by
pΘ(θ,η) :=∑
e∈Eh
κe
|e|
∫
e
[θ] : [η] ds (|e| := length of the edge e), (11)
4
and κe is a positive constant having the same physical dimension as C (for
instance, for smooth C one could take κe as |C| evaluated at the midpoint
of e). Furthermore, Rh : H1(Th) → Γh is a suitable ‘reduction’ operator,
to be defined case by case.
Remark 1.1. We point out that eliminating γh from system (9), our
scheme is equivalent to the following problem involving only the rotations
and the vertical displacements:
Find (θh, wh) ∈ Θh × Wh :
ah(θh,ηh) + λt−2(∇hwh − Rhθh,∇hvh − Rhηh)
= (g, vh) ∀(ηh, vh) ∈ Θh × Wh.(12)
Remark 1.2. The penalty term pΘ defined in (11) makes the bilinear
form ah(·, ·) of (10) to be coercive on the space Θh. Since Θh will be made
by nonconforming vectorial functions, dropping the term pΘ, or taking κe
“excessively small”, may cause a loss of stability for the scheme at hand.
This occurence is briefly highlighted in Section 2.
1.1. The “nonconforming bubble” element
This is the element presented and analyzed in [6], defined as follows. First,
on a generic triangle T ∈ Th we define:
BNC2 (T ) := Span χ2 , (13)
where χ2 denotes the nonconforming bubble of P2, i.e., the polynomial of
degree 2 vanishing at the two Gauss points of each edge. In barycentric
coordinates this bubble has the expression (for instance),
χ2 = 3(λ21 + λ2
2 + λ23) − 2. (14)
The scheme is then given by the following choices.
• The finite element spaces are
Θh =
η : η|T ∈(
P1(T ) ⊕ BNC2 (T )
)2,
∫
e
[η] ds = 0 ∀e ∈ Eh
, (15)
Wh =
v : v|T ∈ P1(T ) ⊕ BNC2 (T ),
∫
e
[v] ds = 0 ∀e ∈ Eh
, (16)
Γh =
τ : τ |T ∈ P0(T )2 ⊕ ∇BNC2 (T )
. (17)
5
• The reduction operator Rh : H1(Th) → Γh is defined locally by:∫
T
(η − Rhη) dx = 0 ∀T ∈ Th, η ∈ H1(Th), (18)
∫
T
div(η − Rhη) dx = 0 ∀T ∈ Th, η ∈ H1(Th). (19)
1.2. The “conforming bubble” element
This element is also mentioned in [6], and it differs from the previous one
only for the choice of the type of bubble. Define, on a generic triangle
T ∈ Th:
B3(T ) := Span b3 , (20)
where b3 denotes the standard cubic bubble. In barycentric coordinates its
expression is, for instance,
b3 = 27λ1λ2λ3. (21)
This scheme is characterized by the following choices.
• The finite element spaces are
Θh =
η : η|T ∈(
P1(T ) ⊕ B3(T ))2
,
∫
e
[η] ds = 0 ∀e ∈ Eh
, (22)
Wh =
v : v|T ∈ P1(T ) ⊕ B3(T ),
∫
e
[v] ds = 0 ∀e ∈ Eh
, (23)
Γh =
τ : τ |T ∈ P0(T )2 ⊕ ∇B3(T )
. (24)
• The reduction operator Rh : H1(Th) → Γh is the same defined in
(18)-(19).
1.3. The P NC
1− P NC
1− P0 element
This element has been introduced and analyzed in [12], and it is character-
ized by the following choices.
• The finite element spaces are
Θh =
η : η|T ∈ (P1(T ))2,
∫
e
[η] ds = 0 ∀e ∈ Eh
, (25)
Wh =
v : v|T ∈ P1(T ),
∫
e
[v] ds = 0 ∀e ∈ Eh
, (26)
Γh =
τ : τ |T ∈ (P0(T ))2
. (27)
6
• Rh is simply the L2-projection operator onto the piecewise constant
functions (see (18)).
All the elements presented in Sections 1.1–1.3 are first order convergent
for the kinematic variables. More precisely, for each method it holds (see [6]
and [12])
||θ − θh||1,h + ||w − wh||1,h ≤ Ch, (28)
where ||·||1,h denotes the H1-broken norm. Furthermore, it has been proved
in [9] that for the element of Section 1.3 one has the L2 bounds:
||θ − θh||0 + ||w − wh||0 ≤ Ch2. (29)
2. Numerical results
In this section we present some numerical results showing the behavior of
the nonconforming elements of Sections 1.1–1.3. For comparison purposes,
we consider also the stable and first order convergent Arnold-Falk element,
proposed and analyzed in [2]. As a test problem we take an isotropic and
homogeneous plate Ω = (0, 1) × (0, 1), clamped on the whole boundary,
for which the analytical solution is explicitly known (see [8] or [9] for all
the details). We analyze the convergence properties of the elements by
considering different uniform decompositions as shown in Figure 1(left)
with h = 1/4, 1/8, 1/16, 1/32, and keeping the thickness sufficiently small
(t = .001). Moreover, in the penalty term (11) we set κe = |C| ' euclidean
norm of C.
In Figures 2 and 3 we report the relative errors, in the L2-norm, for
rotations and deflection respectively.
Instead, Figures 4 and 5 show the relative errors for rotations and
deflection, respectively, in the energy norm.
We conclude this Note by briefly commenting on the effect of choosing
a “too small” penalty parameters κe in (11) (cf. Remark 1.2). To this aim,
we consider the same test problem as before, but we use the mesh shown in
Figure 1(right). Figures 6–7 display the profiles of the analytical solution
for the first component of the rotation vector θ1, and for the deflection w,
respectively. Figures 8–9 show the profiles of the corresponding discrete
solutions, obtained by the element of Section 1.3 with the reasonable choice
κe = |C|; we may observe that the profiles are quite accurately captured.
Figures 10–11 show the profiles of the discrete solutions, this time with the
wrong choice κe = |C| · 10−5; undesirable oscillations clearly appear on the
approximation of both the rotations and the vertical displacements.
7
(0,0) (1,0)
(1,1)(0,1)
(0,0) (1,0)
(1,1)(0,1)
Figure 1. Adopted meshes.
100 101 10210−3
10−2
10−1
100
mesh parameter:1/h
Eθ
Relative rotation errors
12
P1NC−P
1NC−P0 element
conforming bubble elementnonconforming bubble elementArnold−Falk element
Figure 2. Relative rotation errors in L2-norm versus 1/h.
References
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8
100 101 10210−3
10−2
10−1
100
mesh parameter:1/h
Ew
Relative deflection errors
12
P1NC−P
1NC−P0 element
conforming bubble elementnonconforming bubble elementArnold−Falk element
Figure 3. Relative deflection errors in L2-norm versus 1/h.
100 101 102
10−1
100
mesh parameter:1/h
Eθ1
Relative rotation errors
11
P1NC−P
1NC−P0 element
conforming bubble elementnonconforming bubble elementArnold−Falk element
Figure 4. Relative rotation errors in the energy norm versus 1/h.
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100 101 10210−2
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11
P1NC−P
1NC−P0 element
conforming bubble elementnonconforming bubble elementArnold−Falk element
Figure 5. Relative deflection errors in the energy norm versus 1/h.
00.2
0.40.6
0.81
00.2
0.40.6
0.81
−4
−2
0
2
4
x 10−4
analytical θ1
−2.5
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0.81
00.2
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0.810
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0.6
0.8
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8x 10−5
Figure 7. Profile of the analytical solution w.
00.2
0.40.6
0.81
00.2
0.40.6
0.81
−4
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0
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00.2
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7
8x 10−5
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00.2
0.40.6
0.81
00.2
0.40.6
0.81
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−0.01
0
0.01
0.02
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e=|C|10−5
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−0.01
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Figure 10. Profile of the approximation to θ1 with κe = |C| · 10−5.
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Figure 11. Profile of the approximation to w with κe = |C| · 10−5.
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